Munich Personal RePEc Archive
Input aggregation bias in technical
efficiency with multiple criteria analysis
Casasnovas, Valero L. and Aldanondo, Ana M.
Public University of Navarre
12 June 2014
Online at
https://mpra.ub.uni-muenchen.de/56778/
1 Input aggregation bias in technical efficiency with multiple criteria analysis
Ana M. Aldanondo a & Valero L. Casasnovas a,*
a
Department of Business Administration, Public University of Navarre, Campus Arrosadia, Pamplona, 31006, Spain
* Corresponding author. E-mail: [email protected]
WORKING PAPER
June 2014
Abstract
We extend the Tauer (2001) and Färe et al. (2004) analyses of aggregation bias in
technical efficiency measurement to multiple criteria decision analysis. We show input
aggregation conditions consistent with multiple criteria evaluation of overall efficiency
in conjunction with variation in aggregation bias.
Keywords: Data Envelopment Analysis, Input aggregation, multiple objectives.
2 I. Introduction
Aggregate input values are a common feature in Data Envelopment Analysis (DEA)
applications. Researchers estimating the technical efficiency of different economic units
often use raw material costs rather than quantities per material and labour costs instead
of the number of hours worked per job category. While there are advantages in using
aggregate data to overcome information gaps or reduce the dimensions of the problem,
there are also some drawbacks. Various studies (Primont, 1993; Thomas and Tauer,
1994; Tauer, 2001; Färe and Zelenyuk, 2002; Färe et al., 2004) suggest that measures of
radial technical efficiency using linear input aggregates could be biased by price
allocation inefficiency.
The main approach to the theoretical analysis of technical efficiency aggregation bias is
based on the standard model of cost efficiency and its decomposition into technical
efficiency and allocation efficiency. In this context, cost minimization (as a prerequisite
for profit maximization) is the only economic aim of the production unit. The question
is whether this scenario is always warranted. Indeed, in complex production problems,
decisions require a combination of multiple, conflicting optimization criteria and
efficiency. For instance, a plant manager may seek simultaneously to reduce production
costs, optimize management time and reduce pollutant inputs (Coelli et al., 2007). Thus,
we have multiple overall efficiency criteria and a single technology (or technical
efficiency). This may pose an additional problem of inconsistency when estimating
technical efficiency using aggregate inputs. Technical efficiency estimates based on a
single composite input (or materials,…) price may be incompatible with overall
3 The main purpose of this paper is to extend the analysis of input aggregation bias1 to
situations in which the production units are simultaneously focused on several
objectives. The two main issues we wish to address are a) the number and nature of the
aggregate values required in order to avoid inconsistent estimates; and b) how far the
degree of bias in the efficiency score varies when using several linear aggregates of the
same inputs.
II. Presentation of the different models
Let us consider a situation in which we observe the production activity of k=1,…,K
firms using N inputs, xk= (xk1,xk2,…,xkN) Є RN+ in order to produce M outputs yk= (yk1,
yk2,….,ykM) Є RM+ using the same linear technology. We assume that the technology is
regular and presents variable returns to scale (VRS).2 The production possibility set is
considered defined by the convex combination of these observed activities (Färe et al.,
1994). Here, let us recall the definition of the production input set L(y) as the set of
inputs vectors, x Є RN+ yielding at least output level y Є RM+ (Shepard, 1970; Färe et
al.,1994).
Inputs have their corresponding prices (w1,w2,….,wn) Є RN+, which are assumed to be the
same for all firms. To illustrate the problem,3 let us further suppose that the production
process also generates h=1,…,H pollutant residues. Each unit of input, n, includes a
quantity of different materials (an1,an2,….,anH) Є RH+ some of which are discharged into
nature as pollutant residues.4 The total material h used by firm k is given by the
following equation:
1
Similar analysis is possible for aggregate outputs.
2
The analysis can be extended to constant returns to scale.
3
We use the environmental condition to give meaning to the second set of input coefficients. This can be generalised to other problems
4
4
H
h
x
a
A
knN
n nh h
kN
1
,...,
1=
=
∑
=
(1)
Where anh is the unit coefficient of substance h for input n, which may be null for some
inputs. For the sake of simplicity, we will begin by assuming that the productive activity
generates a single pollutant residue, H=1 allowing us to drop the index h in the notation.
Therefore, an increase in technical efficiency in inputs also implies a decrease in costs
and a gain in overall environmental efficiency. We will now consider how the radial
technical efficiency score (Farrell, 1957) varies with an aggregation ofNˆ ≤ N inputs,
simultaneously using two types of linear aggregates of the same inputs, one
price-weighted, wn, and another weighted by the material flow coefficients, an.
N
N
,…,K
,
k=
x
w
C
N
n
kn n N
k
=
∑
≤
=
ˆ
2
1
ˆ
1
ˆ (2)
and
N N K , … , , = k x
a
A kn
N
n n N
k =
∑
≤=
ˆ 2
1
ˆ
1
ˆ (3)
Starting from previous results obtained by Färe et al. (2004), we will compare the
input-oriented technical efficiency scores estimated for firm k using a single linear aggregate
of inputs (costs or materials, indistinctly) with that estimated using two aggregates
simultaneously. To this end, we will define four efficiency scores for the same
production unit k using the data envelopment analysis (DEA) model of Banker, Charnes
and Cooper (1984), BCC, for technologies with variable returns to scale.
EFi(yk,xk), which is the estimated efficiency score, is calculated without aggregate input
5
(
)
K
k
z
z
N
n
x
x
z
M
m
y
y
z
x
y
EFi
k k k k kn kn k k km km k k k,...,
1
1
,
0
,...,
1
,...,
1
to
subject
min
,
=
=
≥
=
≤
=
≥
=
∑
∑
∑
β
β
(4)The efficiency of unit k using aggregate costs of Nˆ inputs, ( , kNˆ, kNˆ 1,..., kN)
k
C y C x x
EF + is
estimated by the following linear program:
(
)
K
k
z
z
N
N
n
x
x
z
C
C
z
M
m
y
y
z
x
x
C
y
EF
k k k k kn kn k N k N k k k k km km k N N k N k k C,...,
1
1
,
0
,...,
1
ˆ
,...,
1
to
subject
min
,...,
,
,
ˆ ˆ 1 ˆ ˆ=
=
≥
+
=
≤
≤
=
≥
=
∑
∑
∑
∑
+β
β
β
(5)We calculate the efficiency score using aggregate material inputs
(
kN kN kN)
k
A y A x x
EF , ˆ, ˆ+1,..., by the same program as in (5), substitutingCkNˆ withAkNˆ.
We then extend program (5) to include both
N k
C ˆ and
N k
A ˆas inputs, in order to obtain the
6
(
)
K
k
z
z
N
N
n
x
x
z
A
A
z
C
C
z
M
m
y
y
z
x
x
A
C
y
EF
k k k k kn kn k N k N k k k N k N k k k k km km k N N k N k N k k CA,...,
1
1
,
0
,...,
1
ˆ
,...,
1
to
subject
min
,...,
,
,
,
ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ=
=
≥
+
=
≤
≤
≤
=
≥
=
∑
∑
∑
∑
∑
+β
β
β
β
(6)Thus, we have four DEA measures of input-oriented technical efficiency:5 EFi,EFc,
EFA, EFCA. The second and third use a single linear aggregate as the input (the subcost
or sub-pollutant), the fourth uses two aggregates of the same inputs.
III. Technical efficiency biases using multiple aggregates
We will begin by showing that the technical efficiency score is consistent and less
biased when the estimation uses two linear aggregates of the same inputs (costs and
environmental load) than when it uses a single aggregate. For this, we start from
previous results obtained by Färe et al. (2004) for an analysis using a single linear
aggregate. These authors show that, for every observation k (k=1,…,K), it is satisfied
that:
(
)
(
)
(
k k)
N N k N k k C kN k
C
y
, C
EF
y
C
x
x
EFi
y
, x
EF
≤
,
ˆ,
ˆ+1,...,
≤
and (7)
(
)
(
)
(
k k)
N N k N k k A kN k
A
y
, A
EF
y
A
x
x
EFi
y
, x
EF
≤
,
,
+,...,
≤
1 ˆ ˆ
Where CkN and AkN are, respectively, the observed firm´s cost and material use; EFC
(yk,CkN) represents the variable returns to scale (VRS) cost efficiency (Färe and
5
7 Grosskopf, 1985); and EFA (yk,AkN) is the VRS overall environmental efficiency. Then,
when some inputs are linearly aggregated while others are kept disaggregated, the
technical efficiency score given by the BCC model exhibits a downwards bias with
respect to that given by an application using fully unbundled inputs, and this bias is
limited by the allocative efficiency (Färe et al., 2004).
According to these results, a unit that performs well in terms of overall environmental
efficiency can be classed as technically inefficient if the calculations are made using
program (5) with a single aggregate cost variable. For this to occur, the unit needs to be
cost allocation inefficient. Similar mis-measurement can occur with a unit that is overall
cost-efficient if the estimate is computed using only aggregate material inputs.
Applying the same conditions as in equations (7), as will be seen later, it can be proven
that, for any k (k=1,…,K), it is satisfied that:
(
)
(
)
(
k k)
N N
k N k N k k CA N
N k N k k
C
y
C
x
x
EF
y
C
A
x
x
EFi
y
x
EF
,
ˆ,
ˆ+1,...,
≤
,
ˆ,
ˆ,
ˆ+1,...,
≤
,
and (8)
(
k,
kNˆ,
kNˆ 1,...,
N)
CA(
k,
kNˆ,
kNˆ,
kNˆ 1,...,
N)
(
k,
k)
A
y
A
x
x
EF
y
C
A
x
x
EFi
y
x
EF
+≤
+≤
In other words, the technical efficiency score estimated by linear program (6), using two
linear aggregates of the same inputs, one for costs (equation 2) the other for materials
(equation 3) and non-aggregated values of all other variables, is biased downwards in
relation to the estimate given by program (4).The degree of bias is less than or equal to
that which occurs in the measure of technical efficiency using a single linear aggregate,
whether of costs or materials. Therefore the use of both aggregates together cannot
possibly give the inconsistent result of finding a unit overall cost- or
environment-efficient and at the same time technically inenvironment-efficient.
To prove the relationship in equations (8), one only needs to demonstrate, in a way
8 by linear program (4) is a subset of the set of feasible solutions given by linear program
(6).6 Therefore, the smallest optimal solution given by program (4) must be greater than
or equal to that given by program (6). Linear program (6) is, in turn, a more constrained
version of program (5). Thus, the smallest optimal solution given by program (6) will be
no smaller than that given by program (5), which proves the relationship in equations
(8).This result can be generalised to the case using more than two linear aggregates of
the same inputs, H>1. An increase in the number of linear aggregates of the same inputs
used in the DEA technical efficiency estimation reduces potential aggregation bias.
Next, we will prove that a sufficient condition for zero aggregation bias in technical
efficiency estimated by linear program (6) is that the input of the kth unit should be a
positive linear combination of the inputs of an overall variable return to scale (VRS)
cost-efficient unit, c(yc,xc) (Färe and Grosskopf, 1985) and an overall VRS
environmentally-efficient unit, a(ya,xa) as indicated by the following expression:7
( )
1, 0
, 0
,..., 1
,..., 1
c
k k
a a
c
an a cn c kn
cm cm km
y L x
N n
x x
x
M m
y y y
∈
≥ + ≥
≥
= +
=
= =
=
α α α
α
α α
(9)
Where index c denotes the overall cost-efficient unit and index a denotes the overall
environmentally-efficient unit (Coelli et al., 2007).
Let us first consider the benchmark case, in which Nˆ = N. That is, where technical
efficiency is estimated by program (6) with only two inputs: the firm’s observed cost
and observed material inputs, EFCA (yk,CkN,,AkN). The purpose of our proof is to
demonstrate that the optimal solution to linear programs (4) and (6) is found with no
6
The proof, omitted for reasons of space and because the procedure is identical to that of Primont (1996) and Färe et al. (2004), is available upon request from the authors. Program (3) has arguably more constraints than program (4) because the input constraint has to be satisfied for all prices.
7
9 slacks in input restrictions for the kth unit; that is, the kth unit is enveloped with a subset
of efficient units of the input set L(yk) (Shepard, 1970; Färe et al., 1983). Here, it can be
recalled that BCC-efficient units (Tone, 1996; Cooper et al., 2007) are those for which
the optimal solution given by the BCC program is equal to 1 with no slacks in input and
output restrictions. Now, we can define as BCC input efficient units those with an
efficiency score equal to unity and no slacks in program (4) input restrictions,8 that is,
no unit in the input set for the output of the BCC input-efficient unit has a smaller input
vector than the BCC input efficient unit (Shepard, 1970; Färe et al., 1983).
We begin with linear program (4). Let it be noted that units, c and a are both
BCC-input-efficient for program (4), because they are run with minimum input costs and
minimum material inputs, respectively, for output yk (Sueyoshi, 1999).9 Furthermore,
the convexity of technology determines that input set L(yk) includes a BCC-input-
efficient unit which has an input vector that is a positive linear combination of the
inputs of the two efficient units, c and a, as indicated by the following expression:10
( )
k a a c a c a c cx x x ∈L y
+ +
+ α α
α α α α λ (10)
where λx≤1.
Therefore, for the kth unit, projected efficiency according to program (4) (z*,y*, x*, β*)
is written as follows:
(
)
K k z z ,...,M m y y z y ,...,N n x x x z x k k k k km km k km an a cn c k kn k n ,..., 1 1 , 0 1 1 * * * * * * * = = ≥ = ≥ = = + = =∑
∑
∑
β
α
α
(11)
8
Tone and Tsutsui (2010) propose an equivalent category in a constant returns to scale model.
9
This condition can be proved by comparing the solutions of the dual of programs (4) and (5). The proof is omitted here but is available upon request from the authors.
10
10 Units c and a, for their part, are BCC input-efficient (Kornbluth, 1974) by linear
program (6); because they have the output yk minimal production cost and lower
environmental impact, respectively. Therefore, as for linear program (4), the projected
activity of linear program (6) (z**, y**, C**, A**, β**) is written as follows:
(
)
(
)
K k z z M m y y z y A A A z A C C C z C k k k km km k k km aN a cN c kN k k kN aN a cN c kN k k kN ,..., 1 1 , 0 ,..., 1 * * * * * * * * * * * * * * * * * * * * = = ≥ = ≥ = + = = + = =∑
∑
∑
∑
α
α
β
α
α
β
(12)Given that (z*,x*) is that unit of the input set which maximally proportionally contracts
the inputs of the kth unit, it is satisfied that β*=β**.11
Finally, it can be shown, in a way analogous to that used by Färe et al. (2004), that the
following relationship is satisfied:
)
,...,
,
,
,
(
)
,
,
(
kNˆ kNˆ kNˆ 1 Nk CA kN
kN k
CA
y
C
A
EF
y
C
A
x
x
EF
≤
+ (13)It only has to be proved that the set of feasible solutions to program (6) for the case in
which Nˆ < Nis a subset of the set of feasible solutions to program (6) whenNˆ = N.
Then, if equations (11), (12) and (13) are satisfied, it is proved that there is zero
aggregate bias in program (6) when the unit under analysis is a positive linear
combination of a cost-efficient unit and an environmentally-efficient unit.
These results can be generalised to the case in which more than two linear aggregates of
the same inputs are used.
The overall conclusion of this study is that aggregate bias in technical efficiency
estimation decreases with the used number of linear aggregates of the same inputs and
is null for allocation-efficient units (in terms of prices, environment,…) and for units
11
11 that are a positive combination of a number of overall efficient units. These results have
important implications for applied analysis. First, DEA technical efficiency programs
with composite inputs should use as many linear aggregates of inputs as overall
efficiency criteria to avoid inconsistencies between technical efficiency and overall
efficiency estimates. Second, an adequate use of multiple linear aggregates could
significantly reduce technical efficiency aggregation bias.
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